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 \begin{Large}

 \noindent Mathematisches Institut  

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 \noindent Universit"at M"unchen
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 \end{Large}

 \vspace*{0.2cm}

 \noindent Mark Hamilton  
 
 \noindent Office hour: Mon 1-2 pm, Room 308 \hspace*{5cm} SS 04
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 {\bf \large Geometry of Manifolds II (Prof. Kotschick) \\ \medskip Problem sheet 3}

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 \noindent 1. Let $G\times M \rightarrow M$ be a smooth group action and $p,q \in M$ points on the same orbit. Show that the stabilizers $G_p$ and $G_q$ are conjugate as subgroups of $G$ (and hence isomorphic). Find an example where $G_p \neq G_q$.  
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 \medskip

 \noindent 2. a) Find an example of a smooth $S^1$-action whose orbits are not all diffeomorphic. 

b) Consider the unitary group $U(n) = \{A \in Mat(n\times n,\mathbb{C})\mid A\cdot \bar{A}^T=I\}$. Show that the standard linear action of $U(n)$ on $\mathbb{C}^n$ induces a transitive action on the unit sphere $S^{2n-1}\subset \mathbb{C}^n$ and determine the stabilizer of a point.

c) Realise the Grassmannian 
\begin{equation*}
Gr(k,n) = \{V \mid V \subset \mathbb{R}^n\,\mbox{$k$-dimensional linear subspace}\}, 
\end{equation*}
where $0\leq k\leq n$, as a homogeneous space. 
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 \medskip

 \noindent 3. Let $\phi:G\times M \rightarrow M$ be a smooth (left) action of a Lie group $G$ on a manifold $M$. The action induces a map 
\begin{equation*}
\phi_*:\mathfrak{g}\rightarrow \mathfrak{X}(M),
\end{equation*}
 given by $(\phi_*X)(p)=\frac{d}{dt}|_{t=0}(exp(tX)\cdot p)$. Show that 
\begin{equation*}
[\phi_*X,\phi_*Y]=-\phi_*[X,Y], \quad \forall X,Y\in \mathfrak{g}.
\end{equation*}
(The minus sign would not be there, if we considered actions from the right.)
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\medskip


 \noindent 4. Let $G\times M \rightarrow M$ be a smooth group action. If $G$ is compact, show that every orbit is an embedded submanifold of $M$. 
\bigskip

 \noindent Please hand in your solutions on Thursday, the 13th of May, 
 in the lecture. 




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